When designing a graphics system, an object should be thought of in mathematical terms rather than in functional term. Therefore it is necessary to define a coordinate system in which the object can be defined analytically.
However our interest should be on object itself rather than on its relationship with some arbitrary coordinate system, i.e. the methods we develop must be coordinate-free or coordinate-independent (See [] for more details ).
Let us denote points as elements of 3-dimensional Euclidean
(or affine) space 
. A point specifies a location
often relative to other objects.
Similarly, vectors can be
considered as elements of 3-dimensional linear (or vector)
space 
.
For any two points 
(as shown in Figure 1)
there is a unique vector v that points from
a to b, i.e.
v is obtained by subtracting a from b
Whereas, given a vector v, there are infinite number of pairs of points
such that v = b - a.
Let w be an arbitrary vector and a, b be a pair of points, then a+w, b+w is another such pair, since
Points and vectors plays a vital role both for designing hardware and for underlying mathematics. For instance, let d be the direction vector.
describes a point on this line segment. As 
increases, the vector from b goes in the direction of d.
Thus the hardware knows to draw the next segment based on the next endpoint and the direction described by the vector d.
Figure 1. Points and Vectors: Vectors are not affected by translation
Let 
. Then
Vectors in a 2-dimensional vector space are given by
Vectors in a 3-dimensional vector space are given by
In general, vectors in n-dimensional vector space is given by
Vectors have length defined by
A unit vector has length 1. Any non-zero vector can be normalized to unit length by dividing it by its length. Vectors are invariant under translation, that is,
Let 
and 
be vectors.
Then the sum of the vectors is
and the difference is
A vector can be scaled by a constant c.
A vector space consists of a set of vector together with the operations of vector addition and scalar multiplication. An affine space consists a set of points, a derived vector space, and two operations viz. vector addition and scalar multiplication.
As we saw in the Introduction, subtraction of
elements of point space 
yields a vector; whereas, addition of elements of
points is not well defined, since different coordinate system would
produce different solutions [].
However, the weighted sums of points is called as barycentric combination, also called the affine combination is defined for points, where the weights sum to one.
A map 
that maps 
onto itself is called an affine map if it leaves
barycentric combinations invariant.
In a given coordinate system, let x be a point defined
by a coordinate triple. An affine map can now be defined as
, where A is a 
matrix
and v is a vector from vector space 
For example, Identity is given by v=0, the zero vector and A=I, the identity matrix. Every affine map can be composed of scalings, rotations, translations, shears.